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Sportsman, Joseph Scott 2014-12-16T06:18:21Z 2014-12-16T06:18:21Z 10000-01-01 1987 1987 In algebraic graph theory one studies algebraic variants of graphs by forming matrices and groups relating to the graph. One example of this is the distance matrices, $\Gamma\sb{\rm i}$, and their associated groups. In this thesis we introduce the graphs, $\Gamma\sp{\rm (r)}$ defined by $\Gamma\sp{\rm (r)}$ = $\Gamma\sb1$ + $\Gamma\sb2$ + $\cdots$ + $\Gamma\sb{\rm r}$ and their automorphism groups G$\sp{\rm (r)}$. We show that for a tree $\Gamma$, the groups G$\sp{\rm (r)}$ form a tower which is not the case for arbitrary graphs. From this, we give a description of the structure of G$\sp{\rm (r)}$ for trees and completely characterize the trees of a fixed diameter which have minimal group tower length. Also we introduce a new parameter, $\chi$ for trees defined as follows: Let x and y be vertices of $\Gamma$. Partition the remaining vertices into three sets; W(x) = $\{$w$\epsilon$V($\Gamma$): $\partial$(w,x)$$ 0$\}$. It turns out that $\chi$ has nice properties. One theorem we prove is the following: If $\Gamma$ is a tree of diameter greater than 3, and m = min$\{\chi$ + 1, (d/2) $\}$, then G$\sp{\rm (m+1)}$ $\not=$ G, but G$\sp{\rm (r)}$ = G for all r $\leq$ m. Made available in DSpace on 2014-12-16T06:18:21Z (GMT). No. of bitstreams: 1 8803208.pdf: 2627812 bytes, checksum: 59b3c0be0e38c308a2158e2ab4b8ad13 (MD5) Previous issue date: 1987 Embargo set by: Seth Robbins for item 71427 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 92 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. http://hdl.handle.net/2142/71261 (UMI)AAI8803208 Mathematics Automorphism Groups of the Augmented Distance Graphs of Trees text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.